This is a preprint of: “Central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem”, Jaume Llibre, Trends in Computer Science and Information Technology, vol. 6(1), 1–6, 2021. DOI: [10.17352/tcsit.000031] CENTRAL CONFIGURATIONS OF THE CIRCULAR RESTRICTED 4-BODY PROBLEM WITH THREE EQUAL PRIMARIES IN THE COLLINEAR CENTRAL CONFIGURATION OF THE 3-BODY PROBLEM JAUME LLIBRE Abstract. In this paper we classify the central configurations of the circu- lar restricted 4-body problem with three primaries with equal masses at the collinear configuration of the 3-body problem and an infinitisimal mass. 1. Introduction and results The well-known Newtonian n-body problem concerns with the motion of n mass points with positive mass mi moving under their mutual attraction in Rd in accor- dance with Newton’s law of gravitation. The equations of the motion of the n-body problem are : Xn mj (ri − rj ) r̈i = − 3 , 1 ≤ i ≤ n, rij j=1,j6=i where we have taken the unit of time in such a way that the Newtonian gravitational constant be one, and ri ∈ Rd (i = 1, . . . , n) denotes the position vector of the i-body, rij = |ri − rj | is the Euclidean distance between the i-body and the j-body. The solutions of the 2-body problem (also called the Kepler problem) has been completely solved. Unfortunately the solutions for the n-body for n > 2 is still an open problem. For the Newtonian n-body problem the simplest possible motions are such that the configuration formed by the n-bodies is constant up to rotations and scaling, such motions are called the homographic solutions of the n-body problem, and are the unique known explicit solutions of the n-body problem when n > 2. Only some special configurations of particles are allowed in the homographic solutions of the n-body problem, called by Wintner [64] central configurations. Also, central config- urations are of utmost importance when studying bifurcations of the hypersurfaces of constant energy and angular momentum, for more details see Meyer [47] and Smale [60]. More precisely, let m1 r1 + · · · + mn rn M = m1 + · · · + mn , c= , M 2010 Mathematics Subject Classification. 70F07,70F15. Key words and phrases. Central configuration, circular restricted 4-body problem. 1 2 J. LLIBRE be the total mass and the center of masses of the n bodies, respectively. A configuration r = (r1 , . . . , rn ) is called a central configuration if the accelera- tion vectors of the n bodies are proportional to their positions with respect to the center of masses with the same constant λ of proportionality, i.e. n X mj (ri − rj ) (1) 3 = λ(rj − c), 1 ≤ j ≤ n, rij j=1,j6=i where λ is the constant of proportionality. Equations (1) are strongly nonlinear and to find the explicit central configura- tions (r1 , . . . , rn ) in function of the masses m1 , . . . , mn when n > 3 is an unsolved problem. Equations (1) are invariant under rotations, dilatations and translations on the plane. Two central configurations are related if we can pass from one to the other doing some of the mentioned transformations. This relation is of equivalence. When we talk about the number of central configurations we will talk about the number of classes of equivalence of central configurations. There is an extensive literature on the study of central configurations, see for instance Euler [28], Lagrange [37], Hagihara [33], Llibre [40, 41], Meyer [47], Moeckel [49], Moulton [49], Saari [56], Smale [60], . . . , and the papers quoted in these references. In this paper we are interested in the planar central configurations of a circular restricted 4-body problem. Of course, for the central configurations of the 4-body problem there are many partial results, see for instance the papers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 42, 43, 44, 45, 46, 50, 51, 52, 53, 54, 55, 57, 58, 59, 63, 65, 66]. We note that the set of central configurations is invariant under translations, rotations, and homothecies with respect their center of mass. It is said that two central configurations are equivalent if after having the same center of mass (doing a translation if necessary) we can pass from one to the other through a rotation around its common center of mass and a homothecy. This defines a relation of equivalence in the set of central configurations. From now on when we talk about a central configuration, we are talking on a class of central configurations under this relation of equivalence. The objective of the present article is to study the central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. We recall that for the 3-body problem when the three masses are equal there is a unique collinear central configuration, where the mass in the middle equidistant from the other two, of course the equal masses can be permuted in the positions of this configuration. Of course as in any circular restricted problem the objective is to describe the motion of the infinitesimal mass with respect to the primaries. Usually this problem is studied in a rotating system of coordinates where the positions of the primaries remain fixed, see for more details on the restricted problems the book of Szebehely [62]. CENTRAL CONFIGURATIONS OF A CIRCULAR RESTRICTED 4-BODY PROBLEM 3 More precisely, taking the unit of mass equal to the masses of the three pri- maries and since a central configuration is invarinat under rotations and homoth- ecies through its center of mass without loss of generality we can assume that the position vector rj of the three primaries with masses m1 = m2 = m3 = 1 are (2) r1 = (x1 , y1 ) = (−1, 0), r2 = (x2 , y2 ) = (0, 0), r3 = (x3 , y3 ) = (1, 0). We denote the position of the infinitesimal mass m4 = 0 by r4 = (x4 , y4 ) = (x, y). Then our main result is the following one. Figure 1. The six central configurations of the circular restricted 4-body problem with three equal primaries in the collinear central configuration of the 3-body problem. The three primaries are in- dicated with the big circles, and the position of the infinitesimal mass in the corresponding six central configurations is indicated with a small circle. Theorem 1. The circular restricted 4-body problem with three primaries of equal masses m1 = m2 = m3 = 1 with position vectors given in (2), and one infinitessimal mass m4 = 0 with position vector r4 = (x4 , y4 ) = (x, y) have the following six central configurations with r4 = pj for j = 1, . . . , 6 being: (i) p1 = (x, y) = (0, 1.1394282249562009..), where the value of the coordinate y is a root of the polynomial −16 − 48y 2 + 40y 3 − 48y 4 + 120y 5 + 23y 6 + 120y 7 − 75y 8 + 40y 9 − 75y 10 − 25y 12 ; (ii) p2 = (x, y) = (0, −1.1394282249562009..); (iii) p3 = (x, y) = (1.7576799791694022.., 0), where the value of the coordinate y is a root of the polynomial −4 + 5x3 − 12x4 − 10x5 + 5x7 ; (iv) p4 = (x, y) = (0.49466649101736443.., 0), where the value of the coordinate y is a root of the polynomial −4 + 8x2 + 21x3 − 4x4 − 10x5 + 5x7 ; (v) p5 = (x, y) = (−0.49466649101736443.., 0); 4 J. LLIBRE (vi) p6 = (x, y) = (−1.7576799791694022.., 0). See Figure 1. The proof of Theorem 1 in given in the next section. 2. Proof of Theorem 1 From (1) we obtain the following eight equations for the central configurations of the 4-body problem in the plane 4 X mj (xi − xj ) ej = 3 = λ(xj − c1 ), 1 ≤ j ≤ 4, rij j=1,j6=i (3) 4 X mj (yi − yj ) ej+5 = 3 = λ(yj − c2 ), 1 ≤ j ≤ 4, rij j=1,j6=i where c = (c1 , c2 ). Substituting in (3) the expressions (2), m1 = m2 = m3 = 1, m4 = 0 and r4 = (x4 , y4 ) = (x, y), corresponding to our circular restricted 4-body problem these eight equations reduce to 5 e1 = −e3 = + λ = 0, 4 e2 = e5 = e6 = e7 = 0, x 1+x 1−x e4 = −λx − 3/2 − 3/2 + 3/2 = 0, (x2 + y2 ) ((x + 1)2 + y2 ) ((x − 1)2 + y 2 ) ! 1 1 1 e8 = y −λ − 3/2 − 3/2 − 3/2 = 0. (x2 + y 2 ) ((x − 1)2 + y 2 ) ((x + 1)2 + y 2 ) Therefore λ = −5/4, and the position vector of r4 = (x, y) in order to have a central configuration of the circular restricted 4-body problem must be a real solution of the system 5 x 1+x 1−x e4 = x − 3/2 − 3/2 + 3/2 = 0, 4 2 (x + y )2 2 2 ((x + 1) + y ) ((x − 1)2 + y 2 ) ! 5 1 1 1 e8 = y − − − = 0. 4 (x2 + y 2 )3/2 ((x − 1)2 + y 2 ) 3/2 ((x + 1)2 + y 2 ) 3/2 In Figure 2 we have shwon the curves e4 (x, y) = 0 and e8 (x, y) = 0, and in Figure 3 the intersection of these two curves. We see that these two curves intersect in six points inside the rectangle R = {(x, y) ∈ R2 : −2.2 ≤ x ≤ 2.2, −2.2 ≤ y ≤ 2.2}. Computing the coordenates of these six points numerically using the Newton method (see for instance [61]), we get the six points pj which appear in the statement of Theorem 1. Of course we have omitted the three points where are located the the three primaries in the intersections of the two curves e4 (x, y) = 0 and e8 (x, y) = 0, because there really these two curves are not defined. Now we shall prove that these six points obtained numerically really are solutions of the system e4 (x, y) = e8 (x, y) = 0. CENTRAL CONFIGURATIONS OF A CIRCULAR RESTRICTED 4-BODY PROBLEM 5 (a) The curve e4 (x, y) = 0 (b) The curve e8 (x, y) = 0 Figure 2. The curves e4 (x, y) = 0 and e8 (x, y) = 0 in the rectan- gle (x, y) ∈ [−2.2, 2.2] × [−2.2, 2.2]. We note that equations e4 (x, y) = e8 (x, y) = 0 are invariant if we change x by −x, and y by −y, so if (x, y) is a solution of the system e4 (x, y) = e8 (x, y) = 0 also (−x, y), (x, −y) and (−x, −y) are solutions. So in order to prove Theorem 1 we only need to study the solutions of system e4 (x, y) = e8 (x, y) = 0 satisfying x ≥ 0 and y ≥ 0. Moreover, from Figure 3 we see that all the are of the form (x, 0) or (0, y), and since in the the origin (0, 0) there is one primary, we must look only for the solutions (x, 0) or (0, y) with x > 0 and y > 0. First we look for the solutions (0, y) with y > 0, then system e4 (x, y) = e8 (x, y) = 0 reduce to 5y 2y 1 (4) − 3/2 − 2 = 0, 4 2 (y + 1) y or equivalently to 8y 3 = (1 + y 2 )3/2 (−4 + 5y 3 ). Squaring the both sides of the this equation we get the equation −16 − 48y 2 + 40y 3 − 48y 4 + 120y 5 + 23y 6 + 120y 7 − 75y 8 + 40y 9 − 75y 10 − 25y 12 = 0. This polynomial equation has only two real roots 0.7625005146027564.. and 1.1394282249562009.., but only the second root satisfies equation (4). This provides the solution p1 of Theorem 1, and consequently also the solution p2 . Now we look for the solutions (x, 0) with x > 0 of the system e4 (x, y) = e8 (x, y) = 0. For these solutions the system reduce to 5x 1 1−x 1+x (5) − 2+ 3/2 = , 4 x |1 − x| |1 + x|3/2 6 J. LLIBRE Figure 3. In this picture we can see the six intersection points between the two curves e4 (x, y) = 0 and e8 (x, y) = 0 different from the positions of the primaries, which provide the six central configurations of the circular restricted 4-body problem with three primaries of equal masses at the collinear configuration of the 3- body problem and an infinitisimal mass squaring the both sides of the previous equality we obtain 1 5 25x2 1 1 (x − 1)(5x3 − 4) 4 − + + 4 − 4 − = 0. x 2x 16 (x − 1) (x + 1) 2x2 |x − 1|3 Writting this equation with a common denominator, which only vanishes at the positions of the primeries, its numerator equal zero can be written as 8(x − 1)5 x2 (x + 1)4 (5x3 − 4) = |x − 1|3 (16 − 64x2 − 40x3 + 96x4 + 288x5 −39x6 − 112x7 − 84x8 + 160x9 + 150x10 − 40x11 − 100x12 + 25x14 ). Squaring again the both sides of the this equality we get (x − 1)6 (−4 + 5x3 − 12x4 − 10x5 + 5x7 )(−4 + 8x2 − 11x3 − 4x4 − 10x5 + 5x7 ) (−4 + 8x2 + 21x3 − 4x4 − 10x5 + 5x7 )(−4 + 16x2 + 5x3 + 4x4 − 10x5 + 5x7 ) = 0. The real zero x = 1 is not good because it correspond to the position of a pri- mary. The unique real root of the polynomial −4 + 5x3 − 12x4 − 10x5 + 5x7 is 1.7576799791694022.. which also is a zero of equation (5), and consequently pro- vides the central configuration p3 , and by the symmetries of the equations of the central configurations also provides the central configuration p6 . The unique real root of the polynomial −4 + 8x2 + 21x3 − 4x4 − 10x5 + 5x7 is 0.4946664910173645.. which also is a zero of equation (5), and consequently CENTRAL CONFIGURATIONS OF A CIRCULAR RESTRICTED 4-BODY PROBLEM 7 provides the central configuration p4 , and due to the symmetries of the equations of the central configurations also provides the central configuration p5 . The real roots of the polynomials −4 + 8x2 − 11x3 − 4x4 − 10x5 + 5x7 and 4 + 16x2 + 5x3 + 4x4 − 10x5 + 5x7 are not zeros of the equation (5). This completes the proof of Theorem 1. Acknowledgements The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] Albouy A., Symétrie des configurations centrales de quatre corps, C. R. Acad. Sci. Paris, 320 (1995), 217–220. [2] Albouy A., The symmetric central configurations of four equal masses, Hamiltonian dynamics and celestial mechanics (Seattle, WA, 1995), 131–135, Contemp. Math. 198, Amer. Math. Soc., Providence, RI, 1996. [3] Albouy A., Fu Y. and Sun S., Symmetry of planar four body convex central configurations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464 (2008), 1355–1365. [4] Albouy A. and Kaloshin V., Finiteness of central configurations of five bodies in the plane, Ann. of Math. (2) 176 (2012), 535–588. [5] Álvarez-Ramı́rez M., Corbera M., Delgado J. and Llibre J., The number of planar central configurations for the 4-body problem is finite when 3 mass positions are fixed, Proc. Amer. Math. Soc. 133 (2005), no. 2, 529–536. [6] Álvarez-Ramı́rez M. and Delgado J., Central configurations of the symmetric restricted 4- body problem, Celestial Mech. Dynam. Astronom. 87 (2003), no. 4, 371–381. [7] Álvarez-Ramı́rez M. and Llibre J., The symmetric central configurations of the 4–body prob- lem with masses m1 = m2 6= m3 = m4 , Appl. Math. Comp. 219 (2013), 5996–6001. [8] Álvarez-Ramı́rez M. and Llibre J., Hjelmslev quadrilateral central configurations, Physics Letters A 383 (2018), 103–109. [9] Álvarez-Ramı́rez M. and Llibre J., Equilic quadrilateral central configurations, Commun. Nonlinear Sci. Numer. Simul. 78 (2019), 104872, 7 pp. [10] Alvarez-Ramirez A., Santos A.A. and Vidal C., On co-circular central configurations in the four and five body-problem for homogeneous force law, J.Dynam. Differential Equations 25 (2013), no.2, 269–290. [11] Arribas M., Abad A., Elipe A. and Palacios M., Equilibria of the symmetric collinear re- stricted four-body problem with radiation pressure, Astrophys. Space Sci. 361 (2016), no. 2, Paper No. 84, 12 pp. [12] Arenstorf R.F., Central configurations of four bodies with one inferior mass, Cel. Mechanics 28 (1982), 9–15. [13] Barros J.F. and Leandro E.S.G., The set of degenerate central configurations in the planar restricted four-body problem, SIAM Journal on Mathematical Analysis 43 (2011), 634–661. [14] Barros J.F. and Leandro E.S.G., Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem, SIAM Joural on Mathematical Analysis 46 (2014), 1185–1203. [15] Bernat J., Llibre J. and Perez-Chavela E., On the planar central configurations of the 4-body problem with three equal masses, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 16 (2009), 1–13. [16] Chenciner A., Are nonsymmetric balanced configurations of four equal masses virtual or real?, Regul. Chaotic Dyn. 22 (2017), no. 6, 677–687. 8 J. LLIBRE [17] Corbera M., Cors J.M., Llibre J. and Perez-Chavela E., Trapezoid central configurations, Appl. Math. and Comput. 346 (2019), 127–142. [18] Corbera M. and Llibre J., Central configurations of the 4-body problem with masses m1 = m2 > m3 = m4 = m > 0 and m small, Appl. Math. Comput. 246 (2014), 121–147. [19] Corbera M., Cors J.M. and Llibre J., On the central configurations of the planar 1 + 3-body problem, Celestial Mech. Dynam. Astronom. 109 (2011), no. 1, 27–43. [20] Corbera M., Cors J.M. and Roberts, G.E., A four-body convex central configuration with perpendicular diagonals is necessarily a kite, Qual. Theory Dyn. Syst. 17 (2018), no. 2, 367–374. [21] Corbera M., Cors J.M. and Roberts, G.E., Classifying four-body convex central configura- tions. Celestial Mech. Dynam. Astronom. 131 (2019), no. 7, Paper No. 34, 27 pp. [22] Cors J.M. and Roberts G.E., Four-body co-circular central configurations, Nonlinearity 25 (2012), 343–370. [23] Cors J.M., Llibre J. and Ollé M., Central configurations of the planar coorbital satellite problem, Celestial Mech. Dynam. Astronom. 89 (2004), no. 4, 319–342. [24] Deng Y., Li B. and Zhang S., Four-body central configurations with adjacent equal masses, J. Geom. Phys. 114 (2017), 329–335. [25] Deng Y., Li B. and Zhang S., Some notes on four-body co-circular central configurations, J. Math. Anal. Appl. 453 (2017), no. 1, 398–409. [26] Deng C. and Zhang S., Planar symmetric concave central configurations in Newtonian four- body problems, J. Geom. Phys. 83 (2014), 43–52. [27] Érdi B. and Czirják Z., Central configuration of four bodies with an axis of symmetry, Celestial Mech. Dynam. Astronom. 125 (2016), no. 1, 33–70. [28] Euler L., De moto rectilineo trium corporum se mutuo attahentium, Novi Comm. Acad. Sci. Imp. Petrop. 11 (1767), 144–151. [29] Fernandes A.C., Llibre J. and Mello L.F., Convex central configurations of the 4-body problem with two pairs of equal masses, Arch. Rational Mech. Anal. 226 (2017), 303–320. [30] Fernandes A.C., Garcia B.A., Llibre J. and Mello L.F., New central configurations of the (n + 1)-body problem, J. Geom. Phys. 124 (2018), 199–207. [31] Gannaway J.R., Determination of all central configurations in the planar 4-body problem with one inferior mass, Ph. D., Vanderbilt University, Nashville, USA, 1981. [32] Grebenikov E.A., Ikhsanov E.V. and Prokopenya A.N., Numeric-symbolic computations in the study of central configurations in the planar Newtonian four-body problem, Computer algebra in scientific computing, 192–204, Lecture Notes in Comput. Sci. 4194, Springer, Berlin, 2006. [33] Hagihara Y., Celestial Mechanics, vol. 1, MIT Press, Massachusetts, 1970. [34] Hampton M., Co-circular central configurations in the four-body problem, EQUADIFF 2003, 993–998. [35] Hampton M. and Moeckel R., Finiteness of relative equilibria of the four-body problem, Invent. Math. 163 (2006), no.2, 289–312. [36] Hassan M.R., Ullah M.S., Aminul H.M. and Prasad U., Applications of planar Newtonian four-body problem to the central configurations, Appl. Appl. Math. 12 (2017), no. 2, 1088– 1108. [37] Lagrange J.L., Essai surle probléme des trois corps, recueil des pièces qui ont remporté le prix de l’Académie royale des Sciences de Paris,tome IX, 1772, reprinted in Ouvres, Vol.6 (Gauthier-Villars, Paris, 1873), pp 229–324. [38] Leandro E.S.G., On the central configurations of the planar restricted four-body problem, J. Differential Equations 226 (2006), no. 1, 323–351. [39] Llibre J., Posiciones de equilibrio relativo del problema de 4 cuerpos, Publicacions Matemàtiques UAB 3 (1976), 73–88. [40] Llibre J., On the number of central configurations in the n-body problem, Celestial Mech. Dynam. Astronom. 50 (1991), 89–96. [41] Llibre J., On the central configurations of the n-body problem, Appl. Math. Nonlinear Sci. 2 (2017), no. 2, 509–518. [42] Llibre J. and Yuan P., Bicentric quadrilateral central configurations, Appl. Math. Comput. 362 (2019), 124507, 7 pp. [43] Llibre J. and Yuan P., Tangential trapezoid central configurations, to appear in Regul. Chaotic Dyn., (2020). CENTRAL CONFIGURATIONS OF A CIRCULAR RESTRICTED 4-BODY PROBLEM 9 [44] Long Y., Admissible shapes of 4-body non-collinear relative equilibria, Adv. Nonlinear Stud. 3 (2003), no. 4, 495–509. [45] Long Y. and Sun S., Four-Body Central Configurations with some Equal Masses, Arch. Ration. Mech. Anal. 162 (2002), 24–44. [46] MacMillan W.D. and Bartky W., Permanent Configurations in the Problem of Four Bodies, Trans. Amer. Math. Soc. 34 (1932), no. 4, 838–875. [47] Meyer K.R., Bifurcation of a central configuration, Celestial Mech. 40 (1987), 273–282. [48] Miranda C. (1940), Un’osservazione su un teorema di Brouwer (Italian), Boll. Un. Mat. Ital. (2) 3 (1940), 5–7. [49] Moeckel R., On central configurations, Math. Z. 205 (1990), 499–517. [50] Ouyang T. and Xie Z., Collinear central configuration in four-body problem, Celestial Mech. Dynam. Astronom. 93 (2005), no. 1-4, 147–166. [51] Pedersen P., Librationspunkte im restringierten Vierkörperproblem, Danske Vid. Selsk. Math. Fys. 21 (1944), 1–80. [52] Pérez-Chavela E. and Santoprete M., Convex four-body central configurations with some equal masses, Arch. Rational Mech. Anal. 185 (2007), 481–494. [53] Piña E., Computing collinear 4-body problem central configurations with given masse, Dis- crete Contin. Dyn. Syst. 33 (2013), no. 3, 1215–1230. [54] Piña E. and Lonngi P., Central configuration for the planar Newtonian four-body problem, Celest. Mech. Dyn. Astron. 108 (2010), 73–93. [55] Rusu D. and Santoprete M., Bifurcations of central configurations in the four-body problem with some equal masses, SIAM J. Appl. Dyn. Syst. 15 (2016), no. 1, 440–458. [56] Saari D.G., On the role and properties of central configurations, Celestial Mech. 21 (1980), 9–20. [57] Shi J. and Xie Z., Classification of four-body central configurations with three equal masses, J. Math. Anal. Appl. 363 (2010), no. 2, 512–524. [58] Shoaib M., Kashif A.R. and Szücs-Csillik I., On the planar central configurations of rhom- boidal and triangular four- and five-body problems, Astrophys. Space Sci. 362 (2017), no. 10, Paper No. 182, 19 pp. [59] Simó C., Relative equilibrium solutions in the four-body problem, Cel. Mechanics 18 (1978), 165–184. [60] Smale S., Topology and mechanics II: The planar n–body problem, Inventiones math. 11 (1970), 45–64. [61] Stoer J. and Bulirsch R., Introduction to numerical analysis, Springer-Verlag, New York, 1980. [62] Szebehely V., Theory of orbits, The restricted problem of three bodies, Academic Press, New York, 1967. [63] Tang J.L., A study on the central configuration in the Newtonian 4-body problem of celestial mechanics, (Chinese) J. Systems Sci. Math. Sci. 26 (2006), no. 6, 647–650. [64] Wintner A., The Analytical Foundations of Celestial Mechanics, Princeton University Press, 1941. [65] Xie Z., Isosceles trapezoid central configurations of the Newtonian four-body problem, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), no. 3, 665–672. [66] Yoshimi N. and Yoshioka A., 3 + 1-Moulton configuration, SUT J. Math. 54 (2018), no. 2, 173–190. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat